Length Function: If a smooth curve C has the equation

y $=$ f$($x$),$ a $$ x $$ b$,$ let s$($x$)$

be the distance along C from the initial point

P$0($a$,$ f$($a$))$ to the point Q$($x$,$ f$($x$)).$

Then s is a function, called the arc length function, and is given by

s$($x$)=$

x

$1+[$f$\text{'}($t$)]2$

$$dta

Find the arc length function for the curve

y $=2$x$3/2$

with starting point P$0(4,16).$